Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC generally requires shorter keys compared to non-EC cryptography and can provide equivalent or higher levels of security. Elliptic curve digital signature algorithm (ECDSA) offers a variant of the digital signature algorithm (DSA) which uses ECC. ECDSA is widely used in distributed blockchain network. For example, Bitcoin, Ethereum and other digital currencies use this algorithm to sign and verify the legality of transactions.
The field of threshold cryptosystem is a cryptographic technique based on secret sharing technology. The basic idea behind threshold cryptosystem is to divide the key K into n shares (k1, k2, . . . , kn) according to a secret sharing protocol. If any X (t≤X≤n) ki values are known, K can be calculated; if less than any tki values are known, K cannot be calculated due to the lack of information. This method is generally called the (t,n) threshold method. The two secret sharing protocols currently widely used are the Shamir secret sharing protocol (SSP) and the Asmuth-Bloom SSP. The Shamir SSP is based on the Lagrange interpolation formula, while the Asmuth-Bloom SSP is based on the Chinese remainder theorem.
Homomorphic encryption (HE) is a special encryption method, which allows operating on the ciphertext to get an encrypted result. That means the result obtained by directly operating on the ciphertext matches the result of encrypting the operation result of operating on the plaintext. From the perspective of abstract algebra, HE maintains homomorphism. According to the type of operation. HE is generally divided into additive homomorphism, multiplicative homomorphism, subtraction homomorphism, and division homomorphism. Simultaneously satisfying the additive homomorphism and the multiplicative homomorphism means algebraic homomorphism that is full homomorphism. Simultaneously satisfying the four kinds of homomorphism means arithmetic homomorphism. With the current encryption algorithms, the RSA algorithm is a kind of multiplicative homomorphism, the Paillier algorithm is a kind of additive homomorphism, and the Gentry algorithm is a kind of full homomorphism.